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Only a severely restricted class of tensor fields can provide classicalspacetime geometries, namely those that can carry matter field equations thatare predictive, interpretable and quantizable. These three conditions on mattertranslate into three corresponding algebraic conditions on the underlyingtensorial geometry, namely to be hyperbolic, time-orientable andenergy-distinguishing. Lorentzian metrics, on which general relativity and thestandard model of particle physics are built, present just the simplesttensorial spacetime geometry satisfying these conditions. The problem offinding gravitational dynamics---for the general tensorial spacetime geometriessatisfying the above minimum requirements---is reformulated in this paper as asystem of linear partial differential equations, in the sense that theirsolutions yield the actions governing the corresponding spacetime geometry.Thus the search for modified gravitational dynamics is reduced to a clearmathematical task.

Gravitational dynamics for all tensorial spacetimes carrying predictive, interpretable and quantizable matter